This
paper discusses my sculptures in relation to their two most basic geometric
classes and the distinctly contrasting aesthetics of each. Viewers have
sometimes remarked on these aesthetic contrasts, which stem entirely from
differing three-dimensional geometries, and my assumption is that they
will always to some degree be felt, if not necessarily in an immediately
articulate sense.

1.
Introduction

All
my sculptures can be viewed geometrically as either surfaces or objects
with volume. On the other hand they all have in common a signature of continuously
varying curvatures which are always in integral relation to a holistic
design logic. This decision to create sculpture distilled to organic essentials
seems to have always been less conscious than natural, and in any event
is now lost in time. Its fundamental nature is clear. It gave me a vision
and a perseverance immune to distractions. It no doubt ultimately influenced
the development of my neuroanatamy, dictating where dense networks of specialized
synaptic connections would grow.

2.
Surfaces and Volumetric Sculptures

It
should be noted that the works I've referred to as surfaces, since actual
sculptures rather than theoretic conceptions of zero thickness, obviously
must have some volume to physically exist, but their aesthetics primarily
relate to their coherence as surfaces. At first I didn't understand the
nature of this coherence. In time, however, I became aware that I was intuitively
approximating how a soap film would minimize the area between their edge
constraints. More particularly these sculptural surfaces have the opposing
upward and downward curves such a chemical film forms when bounded by an
undulating wire ring. These surfaces are in consequence locally minimal
in the degree to which their opposing curves cancel to yield zero mean
curvature (Figures 1 and 2; [1,
2,
3, and 4]).

Despite
their convolutions the volumetric sculptures I've recently made have a
modular simplicity in using a constructive geometry of columnar tori with
circular cross sections. Such tori graduate toward zero mean curvature
on their inner face, and have positive curvature on their outer face radiating
all their curves in the same direction from any of its apices. My recent
volumetric sculptures all have this duality of curvatures (Figures 3,
4a,
4b, 5a, and
5b;
[4, 5]).

3.
Contrasting Aesthetics

Needless
to say sculptures conceived so differently will also differ in their aesthetic
resonance. Hyperbolically curving surfaces with zero mean curvature have
an irreducibly spare elegance in which three dimensional form has the least
surface area it can, relative to its edge constraints, and still exist
theoretically with zero thickness or physically as molecular film or sculpture.
They suggest a visible revelation of the microscopically hidden scaffolding
of nature in processes such as cellular metabolism. They can also be seen
as a visual analogue to sanity at the limits of moral possibility where
it is taut like a chemical film which can contract no further. However,
the most fundamental aesthetic significance these surfaces inherently possess
lies in their visual evocation of nature's economy. No sophisticated mathematical
understanding is necessary for this fraught significance to be felt in
the mind of the viewer as aesthetic experience.

Figure 1:This ribbon surface would more closely approximate zero mean curvature
were it not for an aesthetic decision to deepen its outward facing concavity.
It is configured as a double helix conforming to the surface of egg. I
first laminated a hollow wooden egg, and then subtracted everything save
the ribbon. Photo: Phillip Geller.

My recent
volumetric sculptures are analogous to the surfaces in also having an aesthetic
dimension of economy. In them it relates to the degree of clarified economy
with which their curvatures enclose volume. Resolving sculptural form to
an economy of continuous cross-sectional circularity is always an inexact
approximation. It entails a constant struggle to visualize and finely coordinate
hand movements guided by both vision and kinesthesia. Particularly when
the form is actually part of a larger, complexly integrated layering of
geometries. In my most recent sculpture any form circular in cross-section
would, for instance, be part of either one of the two tori which intersect
each other forming a cleft around which they also spiral while simultaneously
following a periodically curving path over the surface of four spheres
overlapping in a deployment reflecting the logic of the sculpture's global
geometry (Figures 5a and 5b). Here I should note that sculpture whose geometric
complexity can seem rather absurd when described in words, may nonetheless
have immediate perceptual eloquence for the eye. Complex music can have
similarly immediate eloquence for aural perception. The given in each instance
being the incredible sophistication of our sensory processing in its original
adaptation to the natural world. Being able to perceive the world in patterns
of aesthetic congruence is an incentive to live no doubt given to us over
the course of our evolution because it does have survival value. It is
why artists work, suffering through the migrainous complexities of their
work for the serene transparency to be found on the other side.

Figure 2:This approximate locally minimized surface is a metaphor for the atom.
Arching

ribbons represent
electrons, while the hyperbolic curvatures ensconced within form its nucleus. Photo:
Phillip Geller.

Figure 3:This doubled, self-intersecting torus is a variation of a triply twisted
Möbius band. It also has a trefoil's
triplet of over-under crossings. While by virtue of its self-intersection,
it becomes moreover topologically
equivalent to a figure-8 Klein bottle. Photo: Phillip Geller.

I have
been interested in my recent volumetric sculptures not only from the perspective
of my aspiration to beautifully clarify them, but also in terms of their
biomorphic resonance, particularly in suggesting muscular tissue. When
their toroidal columns intersect forming clefts around which they twist,
a convincing sense of muscles in play is conveyed. Rather than the elegant
spareness of zero mean curvature in the hyperbolic form of a soap film,
these sculptures have the sensuous fullness of fruit and the body's season
of youthful maturity. As purely geometric conceptions these sculptures
perhaps only have that much more potency as evocations of the significance
muscularity has for our survival and the aesthetic frisson the selective
pressure of evolution has genetically invested in human sexuality. In them
the non-specific muscularity of both genders is distilled to pure form,
which segues into an appreciation of our athleticism as well as our sense
of the strength and grace of kindred mammal species. Though without implied
differentiation into male or female, these sculptures are nonetheless visually
resonant with contours which potentially connote sexuality. In the end,
however, these sculptures must ultimately be aesthetically successful as
formal compositions without reference to their biomorphic resonance. In
work whose inspiration is a belief in the modest possibility of formal
beauty nothing less is sufficient.

Figure 4a and
4b:Though quite different in its global
geometry, this sculpture is homeomorphic
in all other respects to the object shown in Figure 3. Photo: Phillip Geller.

Figure
4b

Conclusion

My
sculptures are constructed from their foundations as either surfaces or
objects with volume. Subsequently an integrated layering of elementarily
coherent geometries will form the distinct grammars of the works in particular
motif cycles. The eventual organic complexity which emerges in them is
managed initially by orchestrating it one step at a time. Essential to
ultimate success, however, phase transitions must occur at certain points
leading to a "peak" momentum of spontaneous creativity where everything
feels right and thinking is unhesitatingly translated into knowing action.
This state of conscious and neuromuscular grace has been variously characterized
as "flow", the "effortlessness at the height of effort", the "zone" etc.
It is another phylogenetically ancient potential evolution has given our
species. Its rudiments are seen in all species which play, and in ours
with the appearance of culture it has come to be sought in innumerable
ways.

Figure 5a and
5b:In this sculpture two tori intersect
and traverse a complex pathway in space while
twisting 360 degrees around the axis of their intersection. Photo: Phillip
Geller.

Parsing
sculpture to its geometric grammar is somewhat analogous to viewing a living
organism at the reductive level of biochemistry. While viewing sculpture
aesthetically and in the light of the emotional associations it potentially
evokes is correspondingly like appreciating the intelligence incarnate
in a whole living organism. What underlies this is the enrichment of the
world through the emergent dynamics of living nature's complexity. The
creative expression of our neocortex and older substrates of emotion are
part of this no less than the synchronized flashing of fireflies...we and
they are immersed in life as separate species sharing helices of heredity
with the same cheirality formed by the same four letter chemical alphabet.

Figure 5b

My
human faith as an artist draws strength from the accomplishments of those
who have gone before, some known and many more unknown, but all exemplifying
the native visual intelligence of our species so movingly apparent in the
high naturalism and anonymity of early Paleolithic cave art. Just how is
it that we in so many times and places have created artworks of wonderful
mathematical subtlety, as well as pure mathematics, were we not preadapted
to do so using our phylogenetically ancient capacity for logical perception
coupled with a more recent one for symbolic representation which appeared
with the evolution of language. Think of an early speechless hominid unable
yet to call his or her tribe sapient. Think of this creature mentally mapping
a landscape or picking up a stick. This primate would have possibly had
an evolutionary destiny to dream creative solutions, concentrate meaning
in poetry, and have compassion for all sentient beings.